Multi-Step Word Problems Mathematics Curriculum 5

New York State Common Core
5
Mathematics Curriculum
GRADE
GRADE 5 • MODULE 6
Topic E
Multi-Step Word Problems
5.NF.2, 5.NF.3, 5.NF.6, 5.NF.7c, 5.MD.1, 5.MD.5, 5.G.2
Instructional Days:
5
Coherence -Links from:
G4–M1
Place Value, Rounding, and Algorithms for Addition and Subtraction
G4–M3
Multi-Digit Multiplication and Division
G4–M5
Fraction Equivalence, Ordering, and Operations
G4–M6
Decimal Fractions
G4–M7
Exploring Measurement with Multiplication
G6–M1
Ratios and Unit Rates
G6–M2
Arithmetic Operations Including Division of Fractions
G6–M5
Area, Surface Area, and Volume Problems
-Links to:
Topic E provides an opportunity for students to encounter complex, multi-step problems requiring the
application of the concepts and skills mastered throughout the Grade 5 curriculum. Students use all four
operations with both whole and fractional numbers in varied contexts. The problems in Topic E are designed
to be non-routine problems that require students to persevere in order to solve them.
While wrestling with complexity is an important part of Topic E, the true strength of this topic is derived from
the time allocated for students to construct arguments and critique the reasoning of their classmates. After
students have been given adequate time to ponder and solve the problems, two lessons are devoted to
sharing of approaches and solutions. Students will partner to justify their conclusions, communicate them to
others, and respond to the arguments of their peers.
A Teaching Sequence Towards Mastery of Multi-Step Word Problems
Objective 1: Make sense of complex, multi-step problems and persevere in solving them. Share and
critique peer solutions.
(Lessons 21–25)
Topic E:
Date:
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
Multi-Step Word Problems
1/31/14
6.E.1
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Lesson 21 5•6
NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 21
Objective: Make sense of complex, multi-step problems and persevere in
solving them. Share and critique peer solutions.
Suggested Lesson Structure
Fluency Practice

Concept Development

Student Debrief

Total Time
(8 minutes)
(47 minutes)
(5 minutes)
(60 minutes)
Fluency Practice (8 minutes)
 Change Mixed Numbers to Improper Fractions 5.NF.3
(4 minutes)
 Add Unlike Denominators 5.NF.1
(4 minutes)
Change Mixed Numbers to Improper Fractions
(4 minutes)
Materials: (S) Personal white boards
Note: This fluency activity reviews G5–Module 3 concepts.
T:
(Write
.) How many halves are in 1?
S:
2 halves.
T:
(Write
S:
3 halves.
T:
(Write
T:
(Write 3 + .) Write the answer as a mixed number.
S:
(Write
T:
S:
T:
S:
T:
How many halves are in 1?
2 halves.
How many halves are in 2?
4 halves.
How many halves are in 3?
= + .) What is +
= )
)
Lesson 21:
Date:
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
NOTES ON
LESSONS 21─25:
Lesson Sequence for M6–Topic E:
 Lessons 21–22 use a protocol to
solve problems within teams of four.
The number of problems solved will
vary between teams.
 Lesson 23 uses a protocol to share
and critique student solutions from
Lessons 21–22.
 Lesson 24 resumes the problem
solving begun in Lessons 21–22.
 Lesson 25 uses the protocol from
Lesson 23 to again share and
critique student solutions.
Make sense of complex, multi-step problems and persevere in solving
them. Share and critique peer solutions.
1/31/14
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6.E.3
Lesson 21 5•6
NYS COMMON CORE MATHEMATICS CURRICULUM
S:
6 halves.
T:
(Write
=
+ =
S:
(Write
=
+ = .)
) Write the addition sentence, filling in the missing numerators.
Continue the process for the following possible suggestions:
,
,
,
, and
.
Add Unlike Denominators (4 minutes)
Materials: (S) Personal white boards
Note: This activity reviews content from G5–Module 3.
T:
(Write
S:
(Add.)
) Add the fractions. Simplify the sum, if possible.
Repeat the process for
,
,
Concept Development (47 minutes)
Note: This topic culminates the year with five days dedicated to
problem solving. The problems solved in G5–M6–Lessons 21,
22, and 24 and then shared and critiqued in G5–M6–Lessons 23
and 25 are non-routine and multi-step. The intent is to
encourage students to integrate cross-modular knowledge, to
strategize, and to persevere.
In G5–M6–Lessons 21, 22, and 24, a protocol is suggested to
allow for teams (level-alike or student-selected as per the
teacher’s professional discretion) to work at their own pace
through the nine problems with the understanding that one
group may complete two problems while another group
completes them all.
Problems are handed out one at a time to each team
individually as they complete work on each problem to the best
of their ability. (Notes on an approach to this system are
included in the UDL box to the right.)
There are no Exit Tickets for these lessons, shortening the
Student Debrief. This is to allow more time for problem solving.
The Homework includes one story problem similar to the
problems worked in class, and one brainteaser meant to
provide a fun challenge for families. Student work samples and
a full Debrief are included in G5–M6–Lessons 24–25.
Lesson 21:
Date:
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
NOTES ON
MULTIPLE MEANS OF
ENGAGEMENT:
Students will offer solutions that are
less than perfect. Use your
professional discretion when deciding
whether to move a team forward to
the next problem.
Reasons for persisting:
 Do they need to learn
perseverance? (Will this help them
to be more attentive to detail, to
show their work more effectively, or
to work until they get it right?)
Reasons for moving on:
 Will another return to the same
problem crush their enthusiasm?
 Does the team’s current solution
offer a great share and critique
moment for G5–M6–Lessons 24–
25?
Make sense of complex, multi-step problems and persevere in solving
them. Share and critique peer solutions.
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6.E.4
Lesson 21 5•6
NYS COMMON CORE MATHEMATICS CURRICULUM
Materials: (S) Problem Set
Note: Print the Problem Set single-sided. Cut the problems apart, one problem per half page. As this limits
the work space, consider pasting the smaller papers onto a larger 8½" × 11" sheet.
Process for G5–M6–Lessons 21, 22, and 23: Solving Word Problems in Teams of Four
1. Establish the intention of G5–M6–Lessons 21–25 with teams.
Let students know that over the next five days, they will be working in teams to solve some great problems
and share their solutions with peers. Each team will work at its own pace to solve as many problems as
possible. The object is not to compete with other groups, but for each team to do its personal best.
Introduce this protocol to the students: Think, pair, share, and
complete.
Think: Work independently to begin each problem. Read the
problem through quietly.
Pair: Work together with a partner from within the team to
complete the problem.
Share: Share with the other pair of the team of four, giving
each pair an opportunity to share. (A more in-depth analysis
and share and critique will be explored in G5–M6–Lessons 23
and 25.)
Complete: Return to work following the sharing in order to
incorporate ideas that came from the collaboration. Finalize the
solution.
2. Establish a system for teams to communicate the completion
of a problem.
Throughout the session, circulate and check solutions prior to
giving teams the next problem in the sequence. Celebrate
success when appropriate.
NOTES ON
MULTIPLE MEANS OF
ENGAGEMENT:
For G5–M6–Lessons 23 and 25,
consider reconfiguring students into
new groups of four for a more in-depth
share and critique process. Possible
alternatives to this arrangement are
given below:
 Solve the problems for three days
consecutively. Share and critique
for two days consecutively.
 Solve problems for four days, closing
each session with a share and
critique. Day 5 might be used for a
museum walk.
All materials are housed here in G5–
M6–Lesson 21, so that whatever
structure is chosen, this lesson will be
the home base.
3. Let students know that completed work will be collected,
organized, and analyzed.
To prepare for the share and critique protocol in G5–M6–Lessons 23 and 25, compile student work for the
same problem from various teams. For example, after the first day, all sets of student solutions from Problem
1 would be housed in a dedicated folder as would sets of solutions from Problem 2, and so on. This
organization will allow for efficient re-distribution of solutions as students work with members from different
teams to analyze and critique the solution strategies.
Following this lesson’s Debrief are analyses and possible solution strategies for each of the nine problems.
The problem masters are included at the end of this lesson. The analyses and possible solutions are
positioned after the Debrief to emphasize the fact that students will progress through these problems at
different rates as they work within their groups.
Lesson 21:
Date:
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
Make sense of complex, multi-step problems and persevere in solving
them. Share and critique peer solutions.
1/31/14
This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
6.E.5
Lesson 21 5•6
NYS COMMON CORE MATHEMATICS CURRICULUM
Student Debrief (5 Minutes)
Lesson Objective: Make sense of complex, multi-step problems and persevere in solving them. Share and
critique peer solutions.



If you encountered a difficulty while solving the problem, what strategies did you use to keep going?
What advice would you give a classmate who was having trouble with a difficult problem?
What did you learn about yourself as a problem solver today that will help you to be a better
problem solver tomorrow?
Note: There is no Exit Ticket for this lesson.
Lesson 21:
Date:
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
Make sense of complex, multi-step problems and persevere in solving
them. Share and critique peer solutions.
1/31/14
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Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
6.E.6
NYS COMMON CORE MATHEMATICS CURRICULUM
Lessons 21–23 Analysis and Solutions 5•6
Analysis and Solution Strategies for Problems 1–9
Problem 1: Pierre’s Paper
Pierre folded a square piece of paper vertically to make two rectangles. Each rectangle had a perimeter of 39
inches. How long is each side of the original square? What is the area of the original square? What is the
area of one of the rectangles?
This problem calls on student knowledge of the properties of squares and rectangles as well as their
knowledge of area and perimeter. Understanding the relationships between the lengths of the rectangle’s
sides is the key to solving it.
If students are having difficulty moving forward, the following questions may help them:



How does knowing that this figure is a square help us know about the dimensions of the rectangle?
How are the dimensions of the rectangle related to each other?
What is the unit we are counting?
Think of the rectangle’s shorter side (or longer side) as unit.
Below, Solution A solves for the longer side of the rectangle and uses a more abstract representation of the
thinking, while Solution B solves for the shorter side of the rectangle.
Solution B
Solution A
Lesson 21:
Date:
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
Make sense of complex, multi-step problems and persevere in solving
them. Share and critique peer solutions.
1/31/14
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Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
6.E.7
NYS COMMON CORE MATHEMATICS CURRICULUM
Lessons 21–23 Analysis and Solutions 5•6
Problem 2: Shopping with Elise
Elise saved $184. She bought a scarf, a necklace, and a notebook. After her purchases, she still had $39.50.
The scarf cost three-fifths the cost of the necklace, and the notebook was one-sixth as much as the scarf.
What was the cost of each item? How much more did the necklace cost than the notebook?
This problem is fairly straightforward mathematically. However, students will need to find a common unit for
all three items in order to determine the cost of the notebook. Once this is established, the costs of the other
items may be found easily. Students may attempt to find a solution through fraction multiplication. This
approach may stall when trying to determine the fraction of the money spent on the necklace. The following
may provide scaffolding for students experiencing difficulty:



Which item’s tape should be the longest The shortest
How can we make these units the same size?
Begin with the notebook as 1 unit. If the notebook is 1 sixth the cost of the scarf, then how many
times as much is the scarf’s cost to the cost of the notebook?
Both solutions below begin by finding the amount spent on the three items. While both use the cost of the
notebook as 1 unit, Solution A begins with the necklace and uses the fraction information to subdivide the
other tapes. Solution B uses a multiplicative approach thinking of the scarf’s cost as 6 times as much as the
cost of the notebook.
Solution B
Solution A
Lesson 21:
Date:
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
Make sense of complex, multi-step problems and persevere in solving
them. Share and critique peer solutions.
1/31/14
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Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
6.E.8
NYS COMMON CORE MATHEMATICS CURRICULUM
Lessons 21–23 Analysis and Solutions 5•6
Problem 3: The Hewitt’s Carpet
The Hewitt family is buying carpet for two rooms. The dining room is a square that measures 12 feet on each
side. The den is 9 yards by 5 yards. Mrs. Hewitt has budgeted $2,650 for carpeting both rooms. The green
carpet she is considering costs $42.75 per square yard, and the brown carpet’s price is $4.95 per square foot.
What are the ways she can carpet the rooms and stay within her budget?
While the calculations for solving this problem
are simple multiplication and addition, the
path to finding the appropriate numbers on
which to operate requires a high degree of
organization. Students must attend not only
to finding the various combinations that are
possible, but they must also attend to the units
MP.2 in which the areas and prices are given.
Students may choose to use only one unit of
measure for the areas and prices, or they may
use a combination. The following scaffolds
may support struggling students:




Are the areas expressed in the same
unit? Can we use them as they are or
must we convert?
How might we organize the
information so that we can keep track
of our thinking?
What are the combinations of carpet
that Mrs. Hewitt can choose? Predict
which combination will be the most
expensive? Which the least
expensive? How do you know? How
can that prediction help you to move
forward?
Consider the prices per square yard
and per square foot. Which of these
carpets is the more expensive? How
do you know? How might this
information help you to organize your
thoughts?
Solution A
Solution B
Both of the solutions to the right show good
organization of the calculations used to solve.
Solution A converts the carpet prices to match
the area units of the rooms. Solutions B
converts the dimensions of the rooms to
match the units of the prices.
Lesson 21:
Date:
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
Make sense of complex, multi-step problems and persevere in solving
them. Share and critique peer solutions.
1/31/14
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Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
6.E.9
NYS COMMON CORE MATHEMATICS CURRICULUM
Lessons 21–23 Analysis and Solutions 5•6
Problem 4: AAA Taxi
AAA Taxi charges $1.75 for the first mile and $1.05 for each additional mile. How far could Mrs. Leslie travel
for $20 if she tips the cab driver $2.50?
Students encounter a part–part–whole problem with varying unit size in the AAA Taxi Problem. They must
first consider the cost of the first mile and tip, and then determine how many groups of $1.05 can be made
from the remaining $15.75.
To scaffold, consider the following:



Will all of the $20 be used to pay for the mileage? Why not?
Do all the miles cost the same? How do we account for that in our model?
How would you solve this if all the miles cost the same? What if the tip was the same as the cost for
the miles?
Solution A begins by counting on from the first mile. Solution B chooses to represent the problem with a tape
diagram and divides to find how many units with a value of $1.05 there are once the sum of the tip and first
mile are subtracted from the $20.
Solution A
Solution B
Lesson 21:
Date:
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
Make sense of complex, multi-step problems and persevere in solving
them. Share and critique peer solutions.
1/31/14
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Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
6.E.10
NYS COMMON CORE MATHEMATICS CURRICULUM
Lessons 21–23 Analysis and Solutions 5•6
Problem 5: Pumpkins and Squash
Three pumpkins and two squash weigh 27.5 pounds. Four pumpkins and three squash weigh 37.5 pounds.
Each pumpkin weighs the same as the other pumpkins, and each squash weighs the same as the other squash.
How much does each pumpkin weigh? How much does each squash weigh?
This problem is a departure from the routine problems in most of Grade 5 in that students must unitize two
different variables (1 pumpkin and 1 squash) as a single unit. Once the difference is found between the
quantities, students have several avenues for finding the weights of the individual pumpkin and squash.



Draw the tapes to represent the weights for the two situations. Which tape is longer? How much
longer?
How many more pumpkins are in the second tape? How many more squash?
Outline the difference with a red pen. Can you find this same combination in the rest of the tape?
How many can you find?
Both solutions below use tape diagrams to show that the difference between the two known facts is a
combination of one pumpkin and one squash. Next, they reason that the sum of the weights of a pumpkin
and squash is 10 pounds. From there, they can see two of those pumpkin and squash units in relationship to
the 27.5 pound group. It is clear then that the weight of the pumpkin has to be 7.5 pounds.
Solution A
Solution B
Lesson 21:
Date:
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
Make sense of complex, multi-step problems and persevere in solving
them. Share and critique peer solutions.
1/31/14
This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
6.E.11
NYS COMMON CORE MATHEMATICS CURRICULUM
Lessons 21–23 Analysis and Solutions 5•6
Problem 6: Toy Cars and Trucks
Henry had 20 convertibles and 5 trucks in his miniature car collection. After Henry’s aunt bought him some
more miniature trucks, Henry found that one-fifth of his collection consisted of convertibles. How many trucks
did his aunt buy?
This problem requires students to process a before-and-after scenario. The larger quantity in the before
situation becomes the smaller quantity in the after situation. This change in fractional relationship may be
depicted in various ways. Students should be careful to model only 5 fifths in the after model—1 fifth for the
convertibles and 4 fifths for the trucks. Use the following to scaffold student understanding:




Draw Henry’s convertibles and trucks before his aunt gave him more trucks. Draw the convertibles
and trucks after his aunt gave him more.
What amount stayed the same?
Which is more, the cars or trucks? (Ask for both before and after. Have students simply draw the
bars longer and shorter.)
Refer to the convertibles tape in the after model. Ask, “If this is fifth, what is the whole ”
Solution A combines the before and after models into one tape. The numbering on the top represents the
before while the numbering below represents the after. Solution B also uses fraction division to determine
the whole. Solution C uses a unit approach, with the number of trucks in the beginning as 1 unit.
Solution B
Solution A
Solution C
Lesson 21:
Date:
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
Make sense of complex, multi-step problems and persevere in solving
them. Share and critique peer solutions.
1/31/14
This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
6.E.12
NYS COMMON CORE MATHEMATICS CURRICULUM
Lessons 21–23 Analysis and Solutions 5•6
Problem 7: Pairs of Scouts:
Some girls in a Girl Scout troop are pairing up with some boys in a Boy Scout troop to practice square dancing.
Two-thirds of the girls are paired with three-fifths of the boys. What fraction of the scouts is square dancing?
This problem challenges students to consider what they know about fraction equivalence. The key to this
problem lies in recognizing the need for equal numbers of units. That is, equal numerators must be found!
Once students can visualize that 6 of the girls’ units are the same as 6 of the boys’ units, a fraction of the total
number of units can be found. Scaffold with the following:



We know the same number of girls as boys are dancing. Are these units the same size? How can we
make them the same size?
How can 2 units be the same amount as 3 units? Only if one unit is larger than the other. For
example, 2 yards equals 6 feet if we consider 1 larger unit and a smaller unit.
Make sure that once students make 6 units in each tape for the dancing scouts, they also subdivide
the remaining units in each bar. This will create the 19 total units.
Solution A uses a tape diagram to model the equal amounts and then decompose to make the boy and girls
units equal. Solution B uses an array approach to match up girls and boys.
Solution A
Solution B
Lesson 21:
Date:
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
Make sense of complex, multi-step problems and persevere in solving
them. Share and critique peer solutions.
1/31/14
This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
6.E.13
NYS COMMON CORE MATHEMATICS CURRICULUM
Lessons 21–23 Analysis and Solutions 5•6
Problem 8: Sandra’s Measuring Cups
Sandra is making cookies that require 5 cups of oatmeal. She has only two measuring cups: a one-half cup
and a three-fourths cup. What is the smallest number of scoops that she could make in order to get 5 cups?
Recognizing that using a larger unit will require fewer scoops is the beginning of understanding this problem.
Students may try to name the total using all halves or all fourths, but will find that neither measure can be
used exclusively. Using the larger measure first to scoop as much as possible, then moving to scoop the
remainder with the smaller cup is the more efficient method of solving. To scaffold, ask the following
questions:



Which measuring cup is larger? How does knowing which is larger help you?
Predict which measuring cup will do the job more quickly? How do you know?
How many scoops will it take using just the half-cup measure? How many if only the larger cup is
used? Is it possible to scoop all the oatmeal and fill the three-fourths cup every time?
All three solutions pictured below use the strategy of beginning with the larger cup measure. However,
Solution A uses a unitary approach, decomposing the fourths into a multiple of 3 and a multiple of 2. Solution
B counts on by three-fourths and then by halves. Solution C works at the numerical level to guess and check.
Solution A
Solution B
Solution C
Lesson 21:
Date:
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
Make sense of complex, multi-step problems and persevere in solving
them. Share and critique peer solutions.
1/31/14
This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
6.E.14
NYS COMMON CORE MATHEMATICS CURRICULUM
Lessons 21–23 Analysis and Solutions 5•6
Problem 9: Blue Squares
The dimensions of each successive blue square pictured to the right are half
that of the previous blue square. The lower left blue square measures 6
inches by 6 inches.
a. Find the area of the shaded part.
b. Find the total area of the shaded and unshaded parts.
c. What fraction of the figure is shaded?
There are multiple ways to visualize this graphic, each leading to a different
approach to solving. Students may see that there are 3 identical sets of
graduated squares. Out of these 3 identical sets, only 1 set is shaded.
Students may also do the work to find the fraction of the whole that the smallest shaded square represents
and use an additive approach to finding the shaded area. The shaded area might then be used to find the
total area. In contrast, the fraction that is shaded might be used in conjunction with the total area to name
the area of the shaded parts. Scaffolds could include the following:




Can you find the shaded area of just the first three squares (or L’s)?
Cut the graphic apart into separate L’s or separate squares. What can you say about the fraction
that is shaded in each one?
How long is the side of each shaded square?
What if the little square wasn’t missing What would be the area of the whole square What part of
that whole is missing?
Solution A uses the additive approach mentioned above to find the shaded area, which is multiplied by 3 to
find the total. Solution B works backwards to name the fraction that is shaded, then finds the total area by
using subtraction from a by square’s area. These two pieces of information are then used to find the
area of the shaded region in square inches.
Solution B
Solution A
Lesson 21:
Date:
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
Make sense of complex, multi-step problems and persevere in solving
them. Share and critique peer solutions.
1/31/14
This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
6.E.15
NYS COMMON CORE MATHEMATICS CURRICULUM
Lessons 21─23 Problem Set 5•6
Student______________________________________Team ________________Date __________P1
Pierre’s Paper
Pierre folded a square piece of paper vertically to make two rectangles. Each rectangle had a perimeter of 39
inches. How long is each side of the original square? What is the area of the original square? What is the
area of one of the rectangles?
Student______________________________________Team ________________Date __________P2
Shopping with Elise
Elise saved $184. She bought a scarf, a necklace, and a notebook. After her purchases, she still had $39.50.
The scarf cost three-fifths the cost of the necklace, and the notebook was one-sixth as much as the scarf.
What was the cost of each item? How much more did the necklace cost than the notebook?
Lesson 21:
Date:
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
Make sense of complex, multi-step problems and persevere in solving
them. Share and critique peer solutions.
1/31/14
This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
6.E.16
NYS COMMON CORE MATHEMATICS CURRICULUM
Lessons 21─23 Problem Set 5•6
Student______________________________________Team ________________Date __________P3
The Hewitt’s Carpet
The Hewitt family is buying carpet for two rooms. The dining room is a square that measures 12 feet on each
side. The den is 9 yards by 5 yards. Mrs. Hewitt has budgeted $2,650 for carpeting both rooms. The green
carpet she is considering costs $ .75 per square yard, and the brown carpet’s price is $ .95 per square foot.
What are the ways she can carpet the rooms and stay within her budget?
Student______________________________________Team ________________Date __________P4
AAA Taxi
AAA Taxi charges $1.75 for the first mile and $1.05 for each additional mile. How far could Mrs. Leslie travel
for $20 if she tips the cab driver $2.50?
Lesson 21:
Date:
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
Make sense of complex, multi-step problems and persevere in solving
them. Share and critique peer solutions.
1/31/14
This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
6.E.17
NYS COMMON CORE MATHEMATICS CURRICULUM
Lessons 21─23 Problem Set 5•6
Student______________________________________Team ________________Date __________P5
Pumpkins and Squash
Three pumpkins and two squash weigh 27.5 pounds. Four pumpkins and three squash weigh 37.5 pounds.
Each pumpkin weighs the same as the other pumpkins, and each squash weighs the same as the other
squash. How much does each pumpkin weigh? How much does each squash weigh?
Student______________________________________Team ________________Date __________P6
Toy Cars and Trucks
Henry had 0 convertibles and 5 trucks in his miniature car collection. After Henry’s aunt bought him some
more miniature trucks, Henry found that one-fifth of his collection consisted of convertibles. How many
trucks did his aunt buy?
Lesson 21:
Date:
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
Make sense of complex, multi-step problems and persevere in solving
them. Share and critique peer solutions.
1/31/14
This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
6.E.18
NYS COMMON CORE MATHEMATICS CURRICULUM
Lessons 21─23 Problem Set 5•6
Student______________________________________Team ________________Date __________P7
Pairs of Scouts:
Some girls in a Girl Scout troop are pairing up with some boys in a Boy Scout troop to practice square dancing.
Two-thirds of the girls are paired with three-fifths of the boys. What fraction of the scouts is square dancing?
(Each pair is one Girl Scout and one Boy Scout. The pairs are only from these two troops.)
Student______________________________________Team ________________Date __________P8
Sandra’s Measuring Cups
Sandra is making cookies that require 5 cups of oatmeal. She has only two measuring cups: a one-half cup
and a three-fourths cup. What is the smallest number of scoops that she could make in order to get 5 cups?
Lesson 21:
Date:
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
Make sense of complex, multi-step problems and persevere in solving
them. Share and critique peer solutions.
1/31/14
This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
6.E.19
NYS COMMON CORE MATHEMATICS CURRICULUM
Lessons 21─23 Problem Set 5•6
Student______________________________________Team ________________Date __________P9
Blue Squares
The dimensions of each successive blue square pictured to the right are
half that of the previous blue square. The lower left blue square
measures 6 inches by 6 inches.
a. Find the area of the shaded part.
b. Find the total area of the shaded and unshaded parts.
c. What fraction of the figure is shaded?
Lesson 21:
Date:
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
Make sense of complex, multi-step problems and persevere in solving
them. Share and critique peer solutions.
1/31/14
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Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
6.E.20
NYS COMMON CORE MATHEMATICS CURRICULUM
Name
Lesson 21 Homework 5•6
Date
Sara travels twice as far as Eli when going to camp. Ashley travels as far as Sara and Eli together. Hazel
travels 3 times as far as Sara. In total, all four travel a total of 888 miles to camp. How far do each of them
travel?
Lesson 21:
Date:
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
Make sense of complex, multi-step problems and persevere in solving
them. Share and critique peer solutions.
1/31/14
This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
6.E.21
NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 21 Homework 5•6
The following problem is a brainteaser for your enjoyment. It is intended to encourage working together and
family problem solving fun. It is not a required element of this homework assignment.
A man wants to take a goat, a bag of cabbage, and a wolf over to an island. His boat
will only hold him and one animal or item. If the goat is left with cabbage, he’ll eat
it. If the wolf is left with the goat, he’ll eat it. How can the man transport all three
to the island without anything being eaten?
Lesson 21:
Date:
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
Make sense of complex, multi-step problems and persevere in solving
them. Share and critique peer solutions.
1/31/14
This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
6.E.22
Lesson 22 5•6
NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 22
Objective: Make sense of complex, multi-step problems and persevere in
solving them. Share and critique peer solutions.
Suggested Lesson Structure
Fluency Practice

Concept Development

Student Debrief

Total Time
(10 minutes)
(45 minutes)
(5 minutes)
(60 minutes)
Fluency Practice (10 minutes)
 Multiply 5.NBT.5
(4 minutes)
 Change Improper Fractions to Mixed Numbers 5.NF.3
(3 minutes)
 Add Unlike Fractions 5.NF.1
(3 minutes)
Multiply (4 minutes)
Materials: (S) Personal white boards
Note: This drill reviews year-long fluency standards.
T:
S:
Solve 34 × 24 using the standard algorithm.
(Write 34 × 24 = 816 using the standard algorithm.)
Continue the process for 134 × 24, 46 × 42, 346 × 42, and
768 × 37.
Change Mixed Numbers to Improper Fractions
(3 minutes)
Materials: (S) Personal white boards
Note: This fluency activity reviews G5–Module 3 concepts.
T:
NOTES ON
LESSONS 21─25:
Lesson Sequence for M6–Topic E:
 Lessons 21─22 use a protocol to
solve problems within teams of four.
The number of problems solved will
vary between teams.
 Lesson 23 uses a protocol to share
and critique student solutions from
Lessons 21–22.
 Lesson 24 resumes the problem
solving begun in Lessons 21–22.
 Lesson 25 uses the protocol from
Lesson 23 to again share and
critique student solutions.
(Write 1 + .) Say the sum as a mixed number.
S:
.
T:
(Write
.) How many thirds are in 1?
Lesson 22:
Date:
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
Make sense of complex, multi-step problems and persevere in solving
them. Share and critique peer solutions.
1/31/14
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6.E.23
Lesson 22 5•6
NYS COMMON CORE MATHEMATICS CURRICULUM
S:
3 thirds.
T:
(Beneath
S:
4 thirds.
T:
(Write
T:
(Write 3 + ) Write the sum as a mixed number.
S:
(Write
T:
S:
T:
S:
T:
S:
How many thirds are in 1?
3.
How many thirds are in 2?
6.
How many thirds are in 3?
9.
T:
(Write
, write + .) What’s + ?
= )
)
. Beneath it, write
+ =
) Beneath your mixed number, write the addition sentence,
filling in the missing numbers.
S:
(Beneath
, write + =
.)
Continue the process for the following possible sequence:
,
,
,
,
, and
.
Add Unlike Denominators (3 minutes)
Materials: (S) Personal white boards
Note: This fluency activity reviews content from G5–Module 3.
T:
(Write
S:
(Add.)
) Add the fractions. Simplify the sum, if possible.
Repeat the process for
and
Lesson 22:
Date:
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
Make sense of complex, multi-step problems and persevere in solving
them. Share and critique peer solutions.
1/31/14
This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
6.E.24
Lesson 22 5•6
NYS COMMON CORE MATHEMATICS CURRICULUM
Concept Development (45 minutes)
Materials: (S) G5–M6–Lesson 21 Problem Set
Students continue work through the Problem Set presented in
G5–M6–Lesson 21.
1. Re-establish the intention of G5–M6–Lessons 21─22: to give
students the opportunity to solve challenging, multi-step
problems.
NOTES ON
MULTIPLE MEANS OF
EXPRESSION:
An engaging extension is to offer teams
the opportunity to videotape a solution
strategy to one of the problems. The
videos could be used as part of the
share and critique in G5–M6–Lessons
23 and 25.
2. Remind students of the think, pair, share, and complete
protocol.
After having spent G5–M6–Lesson 21 using the protocol, students may now realize that different teams will
need quiet at different times. You may want to establish a system for lowered voices when necessary.
3. Remind teams of how they advance to the next problem.
Re-establish the way for teams to communicate that they have completed a problem and adjust the system
from the first day if it was flawed.
4. Remind students that completed solutions will be collected, organized, and analyzed.
Student Debrief (5 Minutes)
Lesson Objective: Make sense of complex, multi-step problems and persevere in solving them. Share and
critique peer solutions.
The Student Debrief is intended to invite reflection and active processing of the total lesson experience.




If you encountered a difficulty while solving the problem, what strategies did you use to keep going?
Did you apply what you learned yesterday to today’s problems?
What advice would you give a classmate who was having trouble with a hard problem?
What did you learn about yourself today as a problem solver that will help you to be a better
problem solver tomorrow?
Note: There is no Exit Ticket for this lesson.
Lesson 22:
Date:
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
Make sense of complex, multi-step problems and persevere in solving
them. Share and critique peer solutions.
1/31/14
This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
6.E.25
NYS COMMON CORE MATHEMATICS CURRICULUM
Name
Lesson 22 Homework 5•6
Date
Solve using any method. Show all your thinking.
1. Study this diagram showing all squares. Fill in the table.
Figure
1
2
3
4
5
6
7
8
Area in
Square Feet
1 ft2
9 ft2
1 ft2
Lesson 22:
Date:
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
Make sense of complex, multi-step problems and persevere in solving
them. Share and critique peer solutions.
1/31/14
This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
6.E.26
NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 22 Homework 5•6
The following problem is a brainteaser for your enjoyment. It is intended to encourage working together and
family problem solving fun. It is not a required element of this homework assignment.
Remove 3 matches to leave 3 triangles.
Lesson 22:
Date:
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
Make sense of complex, multi-step problems and persevere in solving
them. Share and critique peer solutions.
1/31/14
This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
6.E.27
Lesson 23 5•6
NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 23
Objective: Make sense of complex, multi-step problems and persevere in
solving them. Share and critique peer solutions.
Suggested Lesson Structure
Fluency Practice

Concept Development

Student Debrief

Total Time
(10 minutes)
(45 minutes)
(5 minutes)
(60 minutes)
Fluency Practice (10 minutes)
 Sprint: Change Mixed Numbers into Improper Fractions 5.NF.3
(10 minutes)
Sprint: Change Mixed Numbers into Improper Fractions (10 minutes)
Materials: (S) Change Mixed Numbers into Improper Fractions Sprint
Note: This Sprint reviews G5–Module 3 concepts.
Concept Development (45 minutes)
Materials: (S) G5–M6–Lesson 21 Problem Set
1. Establish the intention and structure of today’s lesson.
Advise students that today they will revisit their solutions
completed in G5–M6–Lessons 21–22 with a new team of three
who also solved that problem. Depending on the class, consider
doing a whole-group guided example using a simple problem
such as, “Mrs. Peterson harvested 500 apples. She gave 1
seventh to her brother and 2 thirds of the remainder to the
food pantry. How many apples does she have left?”
2. Organize new teams of three.
Based upon an analysis of the solutions, students’ strengths,
weaknesses, and inter-relationships, organize teams of three to
present solutions to the same problem.
Lesson 23:
Date:
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
NOTES ON
LESSONS 21─25:
Lesson Sequence for M6–Topic E:

Lessons 21─22 use a protocol to
solve problems within teams of
four. The number of problems
solved will vary between teams.

Lesson 23 uses a protocol to share
and critique student solutions
from Lessons 21─22.

Lesson 24 resumes the problem
solving begun in Lessons 21─22.

Lesson 25 uses the protocol from
Lesson 23 to again share and
critique student solutions.
Make sense of complex, multi-step problems and persevere in solving
them. Share and critique peer solutions.
1/31/14
This work is licensed under a
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6.E.28
Lesson 23 5•6
NYS COMMON CORE MATHEMATICS CURRICULUM
3. Introduce the following suggested protocol to the students.
(See box to the right.)
Step 1 Student A presents her solution step by step to the
others in the group. (Allow two minutes.)
Step 2 Students B and C discuss and make sense of the solution
while Student A listens without intervening. (Allow two
minutes.)
Step 3 Students B and C each ask one question or share one
thought directly related to the written solution and
explanation. (Allow six minutes or three minutes per
question.) Student A responds and whole-group
dialogue follows.
MP.3
Suggested stems:
 Can you explain why you chose to____?
 What did you mean when you wrote (or said) ___?
 I think you omitted _____.
 It might have been easier to understand your
solution if you ____.
 I would argue that ____.
Step 4 Student A explains to the group what has been learned
from the process and what changes would be made to
the solution, if any. (Allow one minute.)
Step 5 Repeat Steps 1─4 for each student on the team.
A NOTE ON
MULTIPLE MEANS OF
REPRESENTATION:
To clarify the share and critique
protocol for the students, you might
post the process listed step by step.
1.
Student A presents her solution to
the group.
2.
Students B and C analyze and
discuss the solution as Student A
listens.
3.
Students B and C each ask a
question or share a thought about
the solution. Student A responds
first.
4.
Student A explains to the group
what has been learned and
specific changes to improve the
solution.
5.
Repeat the process with Students
B and C.
4. Give students about seven minutes to either revise their solution based on their peers’ input, support a
peer’s revision, or continue work on a problem from the set.
Student Debrief (5 minutes)
Lesson Objective: Make sense of complex, multi-step problems and persevere in solving them. Share and
critique peer solutions.
The Student Debrief is intended to invite reflection and active processing of the total lesson experience.




How did sharing and critiquing each other’s work improve your solution?
What emotions did you experience during the share and critique process? (Follow up with additional
questions based on the responses.) When did you experience nervousness? Annoyance? Surprise?
Confusion? Did those emotions change as you went through the process? Why?
How can we improve our sharing and critiquing process, which we will be using again the day after
tomorrow? (Possibly edit the steps together.)
What did you learn today that will make you a better problem solver tomorrow?
Note: There is no Exit Ticket for this lesson.
Lesson 23:
Date:
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
Make sense of complex, multi-step problems and persevere in solving
them. Share and critique peer solutions.
1/31/14
This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
6.E.29
NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 23:
Date:
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
Lesson 23 Sprint 5•6
Make sense of complex, multi-step problems and persevere in solving
them. Share and critique peer solutions.
1/31/14
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6.E.30
NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 23:
Date:
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
Lesson 23 Sprint 5•6
Make sense of complex, multi-step problems and persevere in solving
them. Share and critique peer solutions.
1/31/14
This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
6.E.31
Lesson 23 Homework 5•6
NYS COMMON CORE MATHEMATICS CURRICULUM
Name
Date
In the diagram, the length of S is the length of T. If S has an area of 368 cm2, find the perimeter of the
figure.
S
T
Lesson 23:
Date:
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
16 cm
Make sense of complex, multi-step problems and persevere in solving
them. Share and critique peer solutions.
1/31/14
This work is licensed under a
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6.E.32
NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 23 Homework 5•6
The following problems are puzzles for your enjoyment. They are intended to encourage working together
and family problem solving fun and are not a required element of this homework assignment.
Take 12 matchsticks arranged in a grid as shown below, and remove 2 matchsticks so 2 squares remain. How
can you do this? Draw the new arrangement.
Moving only 3 matchsticks, make the fish turn around and swim the opposite way. Which matchsticks did
you move? Draw the new shape.
Lesson 23:
Date:
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
Make sense of complex, multi-step problems and persevere in solving
them. Share and critique peer solutions.
1/31/14
This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
6.E.33
Lesson 24 5•6
NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 24
Objective: Make sense of complex, multi-step problems and persevere in
solving them. Share and critique peer solutions.
Suggested Lesson Structure
Fluency Practice

Concept Development

Student Debrief

Total Time
(10 minutes)
(45 minutes)
(5 minutes)
(60 minutes)
Fluency Practice (10 minutes)
 Subtract Unlike Denominators 5.NF.1
(4 minutes)
 Order of Operations 5.OA.1
(3 minutes)
 Multiply by Multiples of 10 5.NBT.2
(3 minutes)
Subtract Unlike Denominators (4 minutes)
NOTES ON
LESSONS 21─25:
Materials: (S) Personal white boards
Note: This drill reviews G5–Module 3 content.
T:
(Write
Lesson Sequence for M6–Topic E:

Lessons 21─22 use a protocol to
solve problems within teams of
four. The number of problems
solved will vary between teams.

Lesson 23 uses a protocol to share
and critique student solutions
from G5–M6–Lessons 21–22.

Lesson 24 resumes the problem
solving begun in Lessons 21–22.

Lesson 25 uses the protocol from
Lesson 23 to again share and
critique student solutions.
) Add the fractions. Simplify the
difference if possible.
S:
(Subtract.)
Repeat the process for
,
, and
.
Order of Operations (3 minutes)
Materials: (S) Personal white boards
Note: This fluency prepares students for today’s lesson.
T:
S:
T:
S:
(Write 12 ÷ 3 + 1.) On your boards, write the complete
number sentence.
(Write 12 ÷ 3 + 1 = 5.)
(Write 12 ÷ (3 + 1).) On your boards, copy the expression.
(Write 12 ÷ (3 + 1).)
Lesson 24:
Date:
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
Make sense of complex, multi-step problems and persevere in solving
them. Share and critique peer solutions.
1/31/14
This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
6.E.34
Lesson 24 5•6
NYS COMMON CORE MATHEMATICS CURRICULUM
T:
S:
Write the complete number sentence, performing the operation inside the parentheses.
(Beneath 12 ÷ (3 + 1) = ____, write 12 ÷ 4 = 3.)
Continue this process with the following possible sequence: 20 – 6 ÷ 2, (20 – 6) ÷ 2, 7 × 4 + 3, and 7 × (4 + 3).
Multiply by Multiples of 10 (3 minutes)
Note: This review fluency drill will help preserve skills students learned and mastered in G5–Module 1 and lay
the groundwork for future concepts.
Materials: (S) Personal white boards
T:
(Write 41 × 10.) Say the complete multiplication
sentence.
S: 41 × 10 = 410.
T: (Write 410 × 2 beside 41 × 10 = 410.) Say the complete multiplication sentence.
S: 410 × 2 = 820.
T: (Write 410 × 20 below 410 × 2 = 820.) Write 410 × 20 as a three-factor multiplication sentence, using
a number bond to factor out 10 from 20.
S: 410 × 10 × 2 = 8,200.
T: Show your board. (Check for accuracy.)
NOTES ON
Direct students to solve using the same method for 32 × 30 and
MULTIPLE MEANS OF
43 × 30.
REPRESENTATION:
Concept Development (45 minutes)
Students continue work progressing through the set of nine
problems presented in G5–M6–Lesson 21.
1. Re-establish the intention of G5–M6–Lessons 21–25 to
give students time and support to solve some great
problems. Remind them that tomorrow will again be
devoted to sharing and critiquing each other’s’ work as
they did in G5–M6–Lesson 23.
2. Remind students of the think, pair, share, and complete
process. Invite students to share ways to make their
workspace more effective and joyful.
3. Remind students that it is not the number of the
problems completed but rather quality of the work that
is of most importance.
4. Remind students that solutions will be collected,
organized, and analyzed.
Lesson 24:
Date:
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
If drawing or modeling is not working
for a team when solving a given
problem, suggest acting it out or
modeling it with concrete materials.
Using small balls of clay can be very
empowering to represent a problem.
NOTES ON
MULTIPLE MEANS OF
EXPRESSION:
As students reflect on their growth as
problem solvers, initiate the
conversation using a personal example,
“At first, when solving the Hewitt’s
Carpet, I felt overwhelmed by all the
information. But, once I made a table,
I relaxed and was able to solve it. I
learned that making a table gave me
the support I needed to persevere.”
Make sense of complex, multi-step problems and persevere in solving
them. Share and critique peer solutions.
1/31/14
This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
6.E.35
Lesson 24 5•6
NYS COMMON CORE MATHEMATICS CURRICULUM
Student Debrief (5 Minutes)
Lesson Objective: Make sense of complex, multi-step problems and persevere in solving them. Share and
critique peer solutions.
The Student Debrief is intended to invite reflection and active processing of the total lesson experience.


Did you apply what you learned yesterday to today’s problems? How?
What did you learn about yourself today as a problem solver that will help you to be a better
problem solver tomorrow?
Note: There is no Exit Ticket for this lesson.
Lesson 24:
Date:
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
Make sense of complex, multi-step problems and persevere in solving
them. Share and critique peer solutions.
1/31/14
This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
6.E.36
NYS COMMON CORE MATHEMATICS CURRICULUM
Name
Lesson 24 Homework 5•6
Date
Pat’s Potato Farm grew 490 pounds of potatoes. Pat delivered of the potatoes to a vegetable stand. The
owner of the vegetable stand delivered of the potatoes he bought to a local grocery store which packaged
half of the potatoes that were delivered into 5-pound bags. How many 5-pound bags did the grocery store
package?
Lesson 24:
Date:
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
Make sense of complex, multi-step problems and persevere in solving
them. Share and critique peer solutions.
1/31/14
This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
6.E.37
NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 24 Homework 5•6
The following problems are for your enjoyment. They are intended to encourage working together and family
problem solving fun. They are not a required element of this homework assignment.
Six matchsticks are arranged into an equilateral triangle. How can you arrange them into 4 equilateral
triangles without breaking or overlapping any of them? Draw the new shape.
Kenny’s dog, Charlie, is really smart! Last week, Charlie buried 7 bones in all. He buried them in 5 straight
lines and put 3 bones in each line. How is this possible? Sketch how Charlie buried the bones.
Lesson 24:
Date:
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
Make sense of complex, multi-step problems and persevere in solving
them. Share and critique peer solutions.
1/31/14
This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
6.E.38
Lesson 25 5•6
NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 25
Objective: Make sense of complex, multi-step problems and persevere in
solving them. Share and critique peer solutions.
Suggested Lesson Structure
Fluency Practice

Concept Development

Student Debrief

Total Time
(11 minutes)
(44 minutes)
(5 minutes)
(60 minutes)
Fluency Practice (11 minutes)
 Multiply 5.NBT.5
(4 minutes)
 Order of Operations 5.OA.1
(3 minutes)
 Subtract Unlike Denominators 5.NF.1
(4 minutes)
Multiply (4 minutes)
Materials: (S) Personal white boards
Note: This drill reviews year-long fluency standards.
T:
S:
T:
S:
(Write 4 tens 9 ones × 4 ten 3 ones = __ × __.) Write
the multiplication sentence in standard form.
(Write 49 × 43.)
Solve 49 × 43 using the standard algorithm.
(Write 49 × 43 = 2,107 using the standard algorithm.)
Continue the process for 249 × 43, 67 × 32, 867 × 32, and
938 × 27.
Order of Operations (3 minutes)
Materials: (S) Personal white boards
NOTES ON
LESSONS 21─25:
Lesson Sequence for M6–Topic E:

Lessons 21─22 use a protocol to
solve problems within teams of
four. The number of problems
solved will vary between teams.

Lesson 23 uses a protocol to share
and critique student solutions
from Lessons 21–22.

Lesson 24 resumes the problem
solving begun inLessons 21–22.

Lesson 25 uses the protocol from
Lesson 23 to again share and
critique student solutions.
Note: This fluency prepares students for today’s lesson.
T:
S:
T:
(Write 24 ÷ 3 + 1.) On your boards, write the complete number sentence.
(Write 24 ÷ 3 + 1 = 9.)
(Write 24 ÷ (3 + 1).) On your boards, copy the expression.
Lesson 25:
Date:
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
Make sense of complex, multi-step problems and persevere in solving
them. Share and critique peer responses.
1/31/14
This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
6.E.39
Lesson 25 5•6
NYS COMMON CORE MATHEMATICS CURRICULUM
S:
T:
S:
(Write 24 ÷ (3 + 1).)
Write the complete number sentence, performing the operation inside the parentheses.
(Beneath 24 ÷ (3 + 1) = ____, write 24 ÷ 4 = 6.)
Continue this process with the following possible sequence: 5 × 4 – 2, 5 × (4 – 2), 36 ÷ 6 – 2, and 36 ÷ (6 – 2).
Subtract Unlike Denominators (4 minutes)
Materials: (S) Personal white boards
Note: This drill reviews G5–Module 3 content.
T:
(Write
S:
(Subtract.)
.) Add the fractions. Simplify the difference, if possible.
Repeat the process for
,
, and
.
Concept Development (44 minutes)
Materials: (S) Student work from G5–M6–Lessons 21, 22, and
24
1. Establish the intention and structure of today’s lesson: to
construct arguments, share, and critique peer solutions.
Advise students that today, they will revisit their solutions
completed in G5–M6–Lessons 21, 22, and 24 and discuss their
answers with students who also solved that problem.
A NOTE ON
MULTIPLE MEANS OF
REPRESENTATION:
To clarify the share and critique
protocol for the students, you might
post the process listed step by step.
1. Student A presents her solution to
the group.
2. Students B and C analyze and
discuss the solution as Student A
listens.
2. Re-organize new teams of three (or keep those from G5–M6–
Lesson 23) based upon an analysis of the solutions, students’
strengths, weaknesses, and inter-relationships.
3. Students B and C each ask a
question or share a thought about
the solution. Student A responds
first.
3. Re-introduce the protocol to the students, which may have
been edited during the Debrief of G5–M6–Lesson 23. (See box
to the right.)
4. Student A explains to the group
what has been learned and specific
changes to improve the solution.
Step 1 Student A presents his/her solution step by step to the
others in the group. (Allow two minutes.)
5. Repeat the process with Students B
and C.
Step 2 Students B and C discuss and make sense of the solution while Student A listens without intervening.
(Allow two minutes.)
Step 3 Students B and C each ask one question or share one thought directly related to the written solution
and explanation. (Allow six minutes or three minutes per question.) Student A responds and wholegroup dialogue follows.
Lesson 25:
Date:
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
Make sense of complex, multi-step problems and persevere in solving
them. Share and critique peer responses.
1/31/14
This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
6.E.40
Lesson 25 5•6
NYS COMMON CORE MATHEMATICS CURRICULUM
Suggested stems:





Can you explain why you chose to____?
What did you mean when you wrote (or said)
___?
I think you omitted _____.
It might have been easier to understand your
solution if you ____.
I would argue that ____.
NOTES ON
MULTIPLE MEANS OF
EXPRESSION:
One way to have shy students share
solution strategies or critique is
through the use of puppets. Have the
students put hand puppets as they
explain their solution.
Step 4 Student A explains to the group what has been learned from the process and what changes would be
made to the solution, if any. (Allow one minute.)
Step 5 Repeat Steps ─4 for each student on the team.
4. Give students time to either revise their solution based on their peers’ input or support a peer’s revision.
(7 minutes)
5. File all student solutions in their work portfolio.
Student Debrief (5 minutes)
Lesson Objective: Make sense of complex, multi-step problems and persevere in solving them. Share and
critique peer solutions.
The Student Debrief is intended to invite reflection and active processing of the total lesson experience.





Did your sharing and critiquing experience improve since the last time? How?
What emotions did you experience during the share and critique process? (Follow up with additional
questions based on the responses.) When did you experience nervousness? Annoyance? Surprise?
Confusion?
Did those emotions change as you went through the process? How? Why?
What is the value of seeing other solutions and arguing about ways of solving problems?
What did you learn today that will make you a better problem solver in the future?
Note: There is no Exit Ticket for this lesson.
Lesson 25:
Date:
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
Make sense of complex, multi-step problems and persevere in solving
them. Share and critique peer responses.
1/31/14
This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
6.E.41
NYS COMMON CORE MATHEMATICS CURRICULUM
Name
Lesson 25 Homework 5•6
Date
Fred and Ethyl had 132 flowers altogether at first. After Fred sold of his flowers and Ethyl sold 48 of her
flowers, they had the same number of flowers left. How many flowers did each of them have at first?
Lesson 25:
Date:
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
Make sense of complex, multi-step problems and persevere in solving
them. Share and critique peer responses.
1/31/14
This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
6.E.42
NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 25 Homework 5•6
The following problems are puzzles for your enjoyment. They are intended to encourage working together
and family problem solving fun. They are not a required element of this homework assignment.
Without removing any, move 2 matchsticks to make 4 identical squares. Which matchsticks did you move?
Draw the new shape.
Move 3 matchsticks to form exactly (and only) 3 identical squares. Which matchsticks did you move? Draw
the new shape.
Lesson 25:
Date:
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
Make sense of complex, multi-step problems and persevere in solving
them. Share and critique peer responses.
1/31/14
This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
6.E.43